Navier-Stokes流的形状Optimization
Shape Optimization for Navier-Stokes Flow
摘要：
本文研究了定常非齐次Navier-斯托克斯方程驱动的牛顿粘性不可压缩流体的最优形状设计问题。我们使用三种方法来获得一些给定的成本泛函的形状梯度的结构。第一个是使用Piola变换，并推导出状态导数及其相关的伴随状态;第二个是使用一个包含拉格朗日泛函的极小极大公式的可微性与函数空间参数化技术;最后一个是使用一个极小极大公式的可微性与函数空间嵌入技术。最后，我们将梯度型算法应用到我们的问题中，数值例子表明我们的理论是有用的，所提出的算法是可行的。
英文摘要：
This paper is concerned with the optimal shape design of the newtonian viscous incompressible fluids driven by the stationary nonhomogeneous Navier-Stokes equations. We use three approaches to derive the structures of shape gradients for some given cost functionals. The first one is to use the Piola transformation and derive the state derivative and its associated adjoint state; the second one is to use the differentiability of a minimax formulation involving a Lagrangian functional with a function space parametrization technique; the last one is to employ the differentiability of a minimax formulation with a function space embedding technique. Finally we apply a gradient type algorithm to our problem and numerical examples show that our theory is useful for practical purpose and the proposed algorithm is feasible.
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